Minimizing the symmetric difference distance in conic spline approximation

We show that the complexity (the number of elements) of an optimal parabolic or conic spline approximating a smooth curve with nonvanishing curvature to within symmetric difference distance ε is c1 ε −1/4 + O(1), if the spline consists of parabolic arcs and c2 ε −1/5 +O(1), if it is composed of general conic arcs of varying type. The constants c1 and c2 are expressed in the affine curvature of the curve. We define an equisymmetric conic arc tangent to a curve at its endpoints, to be the (unique) conic such that the areas of the two moons formed by this conic and the given curve are equal, and show that its complexity is asymptotically equal to the complexity of an optimal conic spline. We also show that the symmetric difference distance between a curve and an equisymmetric conic arc tangent at its endpoints is increasing with affine arc length, provided the affine curvature along the arc is monotone. This property yields a simple and an efficient bisection algorithm for the computation of an optimal parabolic or equisymmetric conic spline.